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- #1

\[

f(x) = \lambda\int_0^1xy^2f(y)dy

\]

At order \(\lambda^2\) and \(\lambda^3\), we have repeated zeros so

\[

D(\lambda) = 1 - \frac{\lambda}{4}.

\]

Then we have

\[

\mathcal{D}(x, y;\lambda) = xy^2

\]

so

\[

f(x) = \frac{\lambda}{D(\lambda)}\int_0^1\mathcal{D}(x, y;\lambda)dy.

\]

How do I get the eigenfunction and value from this method?